mod/mod_sqrt.hpp

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• Last update: 2024-09-14 09:20:23+09:00
• Include: #include "mod/mod_sqrt.hpp"

Depends on

• mod/barrett.hpp
• mod/mod_pow.hpp
• mod/mongomery_modint.hpp
• random/base.hpp

Required by

• graph/count/count_labeled_bipartite.hpp
• poly/fps_sqrt.hpp

Verified with

• test/1_mytest/count_bipartite.test.cpp
• test/2_library_checker/number_theory/sqrt_mod.test.cpp
• test/2_library_checker/polynomial/sqrt_of_fps.test.cpp
• test/2_library_checker/polynomial/sqrt_of_fps_sparse.test.cpp

Code

#include "random/base.hpp"

#include "mod/mod_pow.hpp"


// p は素数. 解なしは -1.

int mod_sqrt(int a, int p) {
  if (p == 2) return a;
  if (a == 0) return 0;
  int k = (p - 1) / 2;
  if (mod_pow(a, k, p) != 1) return -1;
  auto find = [&]() -> pi {
    while (1) {
      ll b = RNG(2, p);
      ll D = (b * b - a) % p;
      if (D == 0) return {b, D};
      if (mod_pow(D, k, p) != 1) return {b, D};
    }
  };
  auto [b, D] = find();
  if (D == 0) return b;
  ++k;
  // (b + sqrt(D))^k

  ll f0 = b, f1 = 1, g0 = 1, g1 = 0;
  while (k) {
    if (k & 1) {
      tie(g0, g1) = mp(f0 * g0 + D * f1 % p * g1, f1 * g0 + f0 * g1);
      g0 %= p, g1 %= p;
    }
    tie(f0, f1) = mp(f0 * f0 + D * f1 % p * f1, 2 * f0 * f1);
    f0 %= p, f1 %= p;
    k >>= 1;
  }
  if (g0 < 0) g0 += p;
  return g0;
}

// p は素数. 解なしは -1.

ll mod_sqrt_64(ll a, ll p) {
  if (p == 2) return a;
  if (a == 0) return 0;
  ll k = (p - 1) / 2;
  if (mod_pow_64(a, k, p) != 1) return -1;
  auto find = [&]() -> pair<i128, i128> {
    while (1) {
      i128 b = RNG(2, p);
      i128 D = b * b - a;
      if (D == 0) return {b, D};
      if (mod_pow_64(D, k, p) != 1) return {b, D};
    }
  };
  auto [b, D] = find();
  if (D == 0) return b;
  ++k;
  // (b + sqrt(D))^k

  i128 f0 = b, f1 = 1, g0 = 1, g1 = 0;
  while (k) {
    if (k & 1) {
      tie(g0, g1) = mp(f0 * g0 + D * f1 % p * g1, f1 * g0 + f0 * g1);
      g0 %= p, g1 %= p;
    }
    tie(f0, f1) = mp(f0 * f0 + D * f1 % p * f1, 2 * f0 * f1);
    f0 %= p, f1 %= p;
    k >>= 1;
  }
  return g0;
}

#line 2 "random/base.hpp"

u64 RNG_64() {
  static uint64_t x_
      = uint64_t(chrono::duration_cast<chrono::nanoseconds>(chrono::high_resolution_clock::now().time_since_epoch()).count()) * 10150724397891781847ULL;
  x_ ^= x_ << 7;
  return x_ ^= x_ >> 9;
}

u64 RNG(u64 lim) { return RNG_64() % lim; }

ll RNG(ll l, ll r) { return l + RNG_64() % (r - l); }
#line 2 "mod/mod_pow.hpp"

#line 2 "mod/mongomery_modint.hpp"

// odd mod.
// x の代わりに rx を持つ
template <int id, typename U1, typename U2>
struct Mongomery_modint {
  using mint = Mongomery_modint;
  inline static U1 m, r, n2;
  static constexpr int W = numeric_limits<U1>::digits;

  static void set_mod(U1 mod) {
    assert(mod & 1 && mod <= U1(1) << (W - 2));
    m = mod, n2 = -U2(m) % m, r = m;
    FOR(5) r *= 2 - m * r;
    r = -r;
    assert(r * m == U1(-1));
  }
  static U1 reduce(U2 b) { return (b + U2(U1(b) * r) * m) >> W; }

  U1 x;
  Mongomery_modint() : x(0) {}
  Mongomery_modint(U1 x) : x(reduce(U2(x) * n2)){};
  U1 val() const {
    U1 y = reduce(x);
    return y >= m ? y - m : y;
  }
  mint &operator+=(mint y) {
    x = ((x += y.x) >= m ? x - m : x);
    return *this;
  }
  mint &operator-=(mint y) {
    x -= (x >= y.x ? y.x : y.x - m);
    return *this;
  }
  mint &operator*=(mint y) {
    x = reduce(U2(x) * y.x);
    return *this;
  }
  mint operator+(mint y) const { return mint(*this) += y; }
  mint operator-(mint y) const { return mint(*this) -= y; }
  mint operator*(mint y) const { return mint(*this) *= y; }
  bool operator==(mint y) const {
    return (x >= m ? x - m : x) == (y.x >= m ? y.x - m : y.x);
  }
  bool operator!=(mint y) const { return not operator==(y); }
  mint pow(ll n) const {
    assert(n >= 0);
    mint y = 1, z = *this;
    for (; n; n >>= 1, z *= z)
      if (n & 1) y *= z;
    return y;
  }
};

template <int id>
using Mongomery_modint_32 = Mongomery_modint<id, u32, u64>;
template <int id>
using Mongomery_modint_64 = Mongomery_modint<id, u64, u128>;
#line 2 "mod/barrett.hpp"

// https://github.com/atcoder/ac-library/blob/master/atcoder/internal_math.hpp
struct Barrett {
  u32 m;
  u64 im;
  explicit Barrett(u32 m = 1) : m(m), im(u64(-1) / m + 1) {}
  u32 umod() const { return m; }
  u32 modulo(u64 z) {
    if (m == 1) return 0;
    u64 x = (u64)(((unsigned __int128)(z)*im) >> 64);
    u64 y = x * m;
    return (z - y + (z < y ? m : 0));
  }
  u64 floor(u64 z) {
    if (m == 1) return z;
    u64 x = (u64)(((unsigned __int128)(z)*im) >> 64);
    u64 y = x * m;
    return (z < y ? x - 1 : x);
  }
  pair<u64, u32> divmod(u64 z) {
    if (m == 1) return {z, 0};
    u64 x = (u64)(((unsigned __int128)(z)*im) >> 64);
    u64 y = x * m;
    if (z < y) return {x - 1, z - y + m};
    return {x, z - y};
  }
  u32 mul(u32 a, u32 b) { return modulo(u64(a) * b); }
};

struct Barrett_64 {
  u128 mod, mh, ml;

  explicit Barrett_64(u64 mod = 1) : mod(mod) {
    u128 m = u128(-1) / mod;
    if (m * mod + mod == u128(0)) ++m;
    mh = m >> 64;
    ml = m & u64(-1);
  }

  u64 umod() const { return mod; }

  u64 modulo(u128 x) {
    u128 z = (x & u64(-1)) * ml;
    z = (x & u64(-1)) * mh + (x >> 64) * ml + (z >> 64);
    z = (x >> 64) * mh + (z >> 64);
    x -= z * mod;
    return x < mod ? x : x - mod;
  }

  u64 mul(u64 a, u64 b) { return modulo(u128(a) * b); }
};
#line 5 "mod/mod_pow.hpp"

u32 mod_pow(int a, ll n, int mod) {
  assert(n >= 0);
  a = ((a %= mod) < 0 ? a + mod : a);
  if ((mod & 1) && (mod < (1 << 30))) {
    using mint = Mongomery_modint_32<202311021>;
    mint::set_mod(mod);
    return mint(a).pow(n).val();
  }
  Barrett bt(mod);
  int r = 1;
  while (n) {
    if (n & 1) r = bt.mul(r, a);
    a = bt.mul(a, a), n >>= 1;
  }
  return r;
}

u64 mod_pow_64(ll a, ll n, u64 mod) {
  assert(n >= 0);
  a = ((a %= mod) < 0 ? a + mod : a);
  if ((mod & 1) && (mod < (u64(1) << 62))) {
    using mint = Mongomery_modint_64<202311021>;
    mint::set_mod(mod);
    return mint(a).pow(n).val();
  }
  Barrett_64 bt(mod);
  ll r = 1;
  while (n) {
    if (n & 1) r = bt.mul(r, a);
    a = bt.mul(a, a), n >>= 1;
  }
  return r;
}
#line 3 "mod/mod_sqrt.hpp"

// p は素数. 解なしは -1.

int mod_sqrt(int a, int p) {
  if (p == 2) return a;
  if (a == 0) return 0;
  int k = (p - 1) / 2;
  if (mod_pow(a, k, p) != 1) return -1;
  auto find = [&]() -> pi {
    while (1) {
      ll b = RNG(2, p);
      ll D = (b * b - a) % p;
      if (D == 0) return {b, D};
      if (mod_pow(D, k, p) != 1) return {b, D};
    }
  };
  auto [b, D] = find();
  if (D == 0) return b;
  ++k;
  // (b + sqrt(D))^k

  ll f0 = b, f1 = 1, g0 = 1, g1 = 0;
  while (k) {
    if (k & 1) {
      tie(g0, g1) = mp(f0 * g0 + D * f1 % p * g1, f1 * g0 + f0 * g1);
      g0 %= p, g1 %= p;
    }
    tie(f0, f1) = mp(f0 * f0 + D * f1 % p * f1, 2 * f0 * f1);
    f0 %= p, f1 %= p;
    k >>= 1;
  }
  if (g0 < 0) g0 += p;
  return g0;
}

// p は素数. 解なしは -1.

ll mod_sqrt_64(ll a, ll p) {
  if (p == 2) return a;
  if (a == 0) return 0;
  ll k = (p - 1) / 2;
  if (mod_pow_64(a, k, p) != 1) return -1;
  auto find = [&]() -> pair<i128, i128> {
    while (1) {
      i128 b = RNG(2, p);
      i128 D = b * b - a;
      if (D == 0) return {b, D};
      if (mod_pow_64(D, k, p) != 1) return {b, D};
    }
  };
  auto [b, D] = find();
  if (D == 0) return b;
  ++k;
  // (b + sqrt(D))^k

  i128 f0 = b, f1 = 1, g0 = 1, g1 = 0;
  while (k) {
    if (k & 1) {
      tie(g0, g1) = mp(f0 * g0 + D * f1 % p * g1, f1 * g0 + f0 * g1);
      g0 %= p, g1 %= p;
    }
    tie(f0, f1) = mp(f0 * f0 + D * f1 % p * f1, 2 * f0 * f1);
    f0 %= p, f1 %= p;
    k >>= 1;
  }
  return g0;
}

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